{ 
[Info,A,B:Type]. [X:EClass(A)]. [P:B 
]. [num:A 
]. [init:B].
[f:B A B].
(es-collect-accum(X;x.num[x];init;b,v.f[b;v];b.P[b])
= 
nL.{<fst(nL), list_accum(b,v.f[b;v];init;snd(nL))>}[
Collect(X;x.num[x];L.P[list_accum(b,v.f[b;v];init;L)])]) }
{ Proof }
Definitions occuring in Statement :
es-collect-accum: es-collect-accum(X;x.num[x];init;a,v.f[a; v];a.P[a]),
es-collect: Collect(X;x.num[x];L.P[L]),
es-filter-image: f[X],
eclass: EClass(A[eo; e]),
bool: ,
nat: ,
uall: [x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
lambda: x.A[x],
function: x:A B[x],
pair: <a, b>,
product: x:A 
B[x],
universe: Type,
equal: s = t,
list_accum: list_accum(x,a.f[x; a];y;l),
single-bag: {x}
Definitions :
empty-bag: {},
bag_size_empty: bag_size_empty{bag_size_empty_compseq_tag_def:o},
bag_only_single: bag_only_single{bag_only_single_compseq_tag_def:o}(x),
bag_size_single: bag_size_single{bag_size_single_compseq_tag_def:o}(x),
false: False,
limited-type: LimitedType,
bfalse: ff,
btrue: tt,
eq_bool: p =b q,
lt_int: i <z j,
le_int: i 
z j,
eq_int: (i =
j),
null: null(as),
set_blt: a <
b,
grp_blt: a < b,
infix_ap: x f y,
dcdr-to-bool: [d],
bl-all: (x
L.P[x])_b,
bl-exists: (
xL.P[x])_b,
b-exists: (i<n.P[i])_b,
eq_type: eq_type(T;T'),
qeq: qeq(r;s),
q_less: q_less(r;s),
q_le: q_le(r;s),
deq-member: deq-member(eq;x;L),
deq-disjoint: deq-disjoint(eq;as;bs),
deq-all-disjoint: deq-all-disjoint(eq;ass;bs),
eq_id: a = b,
eq_lnk: a = b,
es-eq-E: e = e',
es-bless: e <loc e',
es-ble: e loc e',
bimplies: p 
q,
band: p 
q,
bor: p 
q,
in-eclass: e X,
bnot: 
b,
int: 
,
unit: Unit,
union: left + right,
implies: P Q,
collect_accm: collect_accm(v.P[v];v.num[v]),
collect_accum: collect_accum(x.num[x];init;a,v.f[a; v];a.P[a]),
collect_filter: collect_filter(),
es-interface-accum: es-interface-accum(f;x;X),
eclass-compose1: f o X,
record-select: r.x,
eq_atom: x =a y,
eq_atom: eq_atom$n(x;y),
decide: case b of inl(x) => s[x] | inr(y) => t[y],
ifthenelse: if b then t else f fi ,
dep-isect: Error :dep-isect,
record+: record+,
bag: bag(T),
void: Void,
prop: 
,
length: ||as||,
natural_number: $n,
subtype: S 
T,
event_ordering: EO,
es-E: E,
event-ordering+: EO+(Info),
top: Top,
list: type List,
assert: 
b,
set: {x:A| B[x]} ,
so_lambda: 
x.t[x],
fpf: a:A fp-> B[a],
strong-subtype: strong-subtype(A;B),
le: A B,
ge: i 
j ,
not: A,
less_than: a < b,
uimplies: b supposing a,
and: P Q,
uiff: uiff(P;Q),
subtype_rel: A r B,
all: x:A. B[x],
axiom: Ax,
es-collect: Collect(X;x.num[x];L.P[L]),
pi2: snd(t),
list_accum: list_accum(x,a.f[x; a];y;l),
pi1: fst(t),
pair: <a, b>,
single-bag: {x},
lambda: x.A[x],
es-filter-image: f[X],
so_apply: x[s1;s2],
apply: f a,
so_apply: x[s],
es-collect-accum: es-collect-accum(X;x.num[x];init;a,v.f[a; v];a.P[a]),
product: x:A B[x],
equal: s = t,
universe: Type,
so_lambda: x y.t[x; y],
eclass: EClass(A[eo; e]),
bool: ,
uall: [x:A]. B[x],
member: t T,
nat: ,
isect: 
x:A. B[x],
function: x:A B[x],
RepUR: Error :RepUR,
skip: Error :skip,
CollapseTHEN: Error :CollapseTHEN,
Auto: Error :Auto,
es-interface-predecessors: (X)(e),
inr: inr x ,
nil: [],
minus: -n,
eclass-val: X(e),
inl: inl x ,
spreadn: spread3,
sqequal: s ~ t,
atom: Atom,
es-base-E: es-base-E(es),
token: "$token",
quotient: x,y:A//B[x; y],
tl: tl(l),
hd: hd(l),
cons: [car / cdr],
append: as @ bs,
es-E-interface: E(X),
es-prior-interface: prior(X),
true: True,
squash: 
T,
es-causl: (e < e'),
real: 
,
grp_car: |g|,
add: n + m,
subtract: n - m,
exists: x:A. B[x],
strongwellfounded: SWellFounded(R[x; y]),
spread: spread def,
has-value: has-value(a),
or: P Q,
eq_knd: a = b,
callbyvalue: callbyvalue,
sq_type: SQType(T),
l_member: (x l),
p-outcome: Outcome,
rationals: 
,
filter: filter(P;l),
qabs: |r|,
es-le: e loc e' ,
es-locl: (e <loc e'),
es-p-le: e p e',
es-causle: e c e',
es-p-locl: e p< e',
causal-predecessor: causal-predecessor(es;p),
record: record(x.T[x]),
atom: Atom$n,
intensional-universe: IType,
fpf-sub: f g,
map-class: (f[v] where v from X),
deq: EqDecider(T),
ma-state: State(ds),
fpf-dom: x dom(f),
guard: {T},
class-program: ClassProgram(T),
fpf-cap: f(x)?z,
map: map(f;as),
es-loc: loc(e),
Id: Id,
isl: isl(x),
is_list_splitting: is_list_splitting(T;L;LL;L2;f),
is_accum_splitting: is_accum_splitting(T;A;L;LL;L2;f;g;x),
req: x = y,
rnonneg: rnonneg(r),
rleq: x y,
i-member: r I,
partitions: partitions(I;p),
modulus-of-ccontinuity: modulus-of-ccontinuity(omega;I;f),
sq_stable: SqStable(P),
decide_bfalse: decide_bfalse{decide_bfalse_compseq_tag_def:o}(v11.g[v11]; v21.f[v21]),
label: ...$L... t,
cand: A c B,
sum-map: 
f[x] for x L,
sum: (f[x] | x < k),
imax: imax(a;b),
multiply: n * m,
IdLnk: IdLnk,
locl: locl(a),
Knd: Knd,
divides: b | a,
assoced: a ~ b,
set_leq: a b,
set_lt: a <p b,
grp_lt: a < b,
l_contains: A B,
reducible: reducible(a),
prime: prime(a),
l_exists: (xL. P[x]),
l_all: (xL.P[x]),
fun-connected: y is f*(x),
qle: r s,
qless: r < s,
q-rel: q-rel(r;x),
sq_exists: x:{A| B[x]},
i-finite: i-finite(I),
i-closed: i-closed(I),
fset-member: a s,
f-subset: xs ys,
fset-closed: (s closed under fs),
l_disjoint: l_disjoint(T;l1;l2),
cs-not-completed: in state s, a has not completed inning i,
cs-archived: by state s, a archived v in inning i,
cs-passed: by state s, a passed inning i without archiving a value,
cs-inning-committed: in state s, inning i has committed v,
cs-inning-committable: in state s, inning i could commit v ,
cs-archive-blocked: in state s, ws' blocks ws from archiving v in inning i,
cs-precondition: state s may consider v in inning i,
existse-before: e<e'.P[e],
existse-le: ee'.P[e],
alle-lt: e<e'.P[e],
alle-le: ee'.P[e],
alle-between1: e[e1,e2).P[e],
existse-between1: e[e1,e2).P[e],
alle-between2: e[e1,e2].P[e],
existse-between2: e[e1,e2].P[e],
existse-between3: e(e1,e2].P[e],
es-fset-loc: i locs(s),
es-r-immediate-pred: es-r-immediate-pred(es;R;e';e),
same-thread: same-thread(es;p;e;e'),
collect-event: collect-event(es;X;n;v.num[v];L.P[L];e),
decidable: Dec(P),
iff: P 
Q,
uni_sat: a = !x:T. Q[x],
inv_funs: InvFuns(A;B;f;g),
inject: Inj(A;B;f),
eqfun_p: IsEqFun(T;eq),
refl: Refl(T;x,y.E[x; y]),
urefl: UniformlyRefl(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
usym: UniformlySym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
utrans: UniformlyTrans(T;x,y.E[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y]),
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y]),
connex: Connex(T;x,y.R[x; y]),
uconnex: uconnex(T; x,y.R[x; y]),
coprime: CoPrime(a,b),
ident: Ident(T;op;id),
assoc: Assoc(T;op),
comm: Comm(T;op),
inverse: Inverse(T;op;id;inv),
bilinear: BiLinear(T;pl;tm),
bilinear_p: IsBilinear(A;B;C;+a;+b;+c;f),
action_p: IsAction(A;x;e;S;f),
dist_1op_2op_lr: Dist1op2opLR(A;1op;2op),
fun_thru_1op: fun_thru_1op(A;B;opa;opb;f),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
cancel: Cancel(T;S;op),
monot: monot(T;x,y.R[x; y];f),
monoid_p: IsMonoid(T;op;id),
group_p: IsGroup(T;op;id;inv),
monoid_hom_p: IsMonHom{M1,M2}(f),
grp_leq: a b,
integ_dom_p: IsIntegDom(r),
prime_ideal_p: IsPrimeIdeal(R;P),
no_repeats: no_repeats(T;l),
value-type: value-type(T),
valueall-type: valueall-type(T)
Lemmas :
union_subtype_base,
list_subtype_base,
isect_subtype_base,
sq_stable__all,
squash_wf,
sq_stable_from_decidable,
decidable__le,
length_wf_nat,
length_nil,
length_wf2,
bfalse_wf,
sq_stable__assert,
isl_wf,
collect_accm-wf2,
subtype_rel_list,
Id_wf,
es-interface-predecessors_wf,
list_accum_append,
es-loc_wf,
intensional-universe_wf,
eclass-val_wf,
eclass-val_wf2,
es-prior-interface-causl,
false_wf,
true_wf,
assert_elim,
list-subtype,
l_member_wf,
list-set-type2,
subtype_base_sq,
bool_subtype_base,
real-has-value,
nat_inc_real,
lt_int_wf,
int_inc_real,
assert_of_lt_int,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
le_int_wf,
ge_wf,
nat_properties,
es-causl-swellfnd,
le_wf,
es-causl_wf,
es-interface-predecessors-step-sq,
es-prior-interface_wf1,
ifthenelse_wf,
append_wf,
es-prior-interface_wf,
es-interface-subtype_rel2,
es-E-interface_wf,
es-base-E_wf,
subtype_rel_self,
nat_wf,
es-filter-image_wf,
assert_wf,
es-collect_wf,
es-collect-accum_wf,
event-ordering+_inc,
es-E_wf,
event-ordering+_wf,
eclass-ext,
single-bag_wf,
list_accum_wf,
pi2_wf,
pi1_wf,
eclass_wf,
bool_wf,
length_wf1,
pi1_wf_top,
top_wf,
member_wf,
bag_wf,
eqtt_to_assert,
not_wf,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
bnot_wf,
in-eclass_wf,
subtype_rel_wf,
es-interface-top,
empty-bag_wf
\mforall{}[Info,A,B:Type]. \mforall{}[X:EClass(A)]. \mforall{}[P:B {}\mrightarrow{} \mBbbB{}]. \mforall{}[num:A {}\mrightarrow{} \mBbbN{}]. \mforall{}[init:B]. \mforall{}[f:B {}\mrightarrow{} A {}\mrightarrow{} B].
(es-collect-accum(X;x.num[x];init;b,v.f[b;v];b.P[b])
= \mlambda{}nL.\{<fst(nL), list\_accum(b,v.f[b;v];init;snd(nL))>\}[
Collect(X;x.num[x];L.P[list\_accum(b,v.f[b;v];init;L)])])
Date html generated:
2011_08_16-PM-05_27_42
Last ObjectModification:
2011_06_20-AM-01_23_59
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